Extending Kato’s result
to elliptic curves with p-isogenies
Christian Wuthrich
13th January 2006
Abstract
Let E be an elliptic curve without complex multiplication defined over Q and let
p be an odd prime number at which E has good and ordinary reduction. Kato has
proved in [Kat04] the first half of the main conjecture for E under the condition thatthe representation ρp : GQ
 Aut(Tp E) of the absolute Galois group of Q attachedto the Tate module Tp E is surjective. We prove here that the result still holds if the Eadmits an isogeny of degree p. As a by-product, we show that the p-adic L-functions
attached to an elliptic curve with good ordinary reduction at p is always an integral
series.11G05, 11G40, 11R23, 11F67

1Introduction

Let E be an elliptic curve without complex multiplication defined over Q and let p > 2
be a prime number. Suppose that E has good ordinary reduction at p. We denote byρp : G Q
 Aut(Tp E) the representation of the absolute Galois group of Q attachedto the Tate module Tp E.
Let Ep∞ be the group of all points on E whose order is a power of p. Let ∞ Q bethe cyclotomic Zp -extension and n Q its n-th layer. The Selmer group of E is definedas the kernel of the mapY 1´`H (n Qυ , E) ,S(E/n Q) = ker H1 (n Q, Ep∞ )υwhere the product runs over all places υ in n Q. The Pontryagin dual of the direct limitof these groups under the restriction maps`
´X(E/∞ Q) = Hom lim S(E/n Q), Qp/Zp−→
has naturally the structure of a finitely generated Λ-module, if Λ denotes the Iwasawa algebra of the Zp -extension ∞ Q/Q. By Theorem 17.4 of [Kat04], we know thatX(E/∞ Q) is Λ-torsion. The characteristic ideal charΛ (X(E/∞ Q)) in Λ is an importantalgebraic object attached to E and p.
On the analytic side, Mazur and Swinnerton-Dyer have constructed in [MSD74] ap-adic L-function Lp (E/Q, T ) in Λ ⊗ Qp . See section 3 for more details. It was conjectured that this series has integral coefficients. We will prove the following extension ofProposition 3.7 in [GV00].
Theorem 5.
The analytic p-adic L-function Lp (E/Q, T ) belongs to Λ for all elliptic curves E/Qwith good ordinary reduction at p > 2.
The conclusion can certainly not be extended to the supersingular case since thep-adic L-functions in this case will never be integral. The supersingular case is well
explained in [Pol03] where it is shown how one can extract integral power series.

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The main conjecture asserts that the element Lp (E/Q, T ) generates the characteristic ideal charΛ (X(E/∞ Q)). Kato has proved in [Kat04] the first half of the main
conjecture under the assumption that the representation ρp is surjective. Our aim isto extend his result to curves where the GQ -module E[p] is reducible.
Theorem 4.
Let E/Q be an elliptic curve without complex multiplication and let p > 2 be a prime.
Suppose that E has good ordinary reduction at p and that the representation ρ p is either
surjective or that E[p] is reducible. Then charΛ (X(E/∞ Q)) divides the ideal generatedby Lp (E/Q, T ).
The same argument does not extend to the remaining cases; for them we only obtaina conditional result. See Proposition 7.
In special cases, Greenberg and Vatsal have proved in [GV00] the full main conjecture. Namely if the E admits an isogeny of degree p whose kernel is either ramified atp and odd or unramified at p and even.
The paper consists of two parts. The first part concerns tha so-called fine Selmer
group. The existence of Kato’s Euler system gives directly a bound on this group. We
use a result of Coates and Sujatha in [CS05] to strengthen the usual bound.
The second part transfers the bound from the fine Selmer group to the Selmer groupusing global duality. The proof of theorem 4 is first done on the so-called optimal curve
where one knows already that the p-adic L-function is integral.
The author wishes to thank John Coates, Robert Pollack, and Karl Rubin for helpful
discussions.

2The fine Selmer group

Let E be an elliptic curve defined over Q and let p be any odd prime. We define
the fine1 Selmer group to be the subgroup of S(E/n Q) defined by imposing stronger
conditions at the completion n Qp of n Q at the unique prime p above p :0 R(E/n Q)

 S(E/n Q)

 H 1 ( n Qp , E p∞ )The dual of the direct limit of the groups R(E/n Q) will be denoted by Y (E/∞ Q); it isagain a finitely generated Λ-module. Theorem 12.4.1 in [Kat04] proves that Y (E/∞ Q)is Λ-torsion. Denote by charΛ (Y (E/∞ Q)) the characteristic ideal of Y (E/∞ Q) in Λ.
Kato constructs an Euler system c attached to E and p. This is a collection ofcohomology classes cK ∈ H 1 (K, Tp E) for sufficiently many abelian extensions K of Q,including K = Q(µ[pk ]) for all k > 0. The norm compatibility imposed on an Euler
system, provides us with an element ∞ c in the projective limit
11∞ c ∈ lim H (n Q, Tp E) = ∞ H (Q, Tp E)←−nwhere the limit follows the corestriction map. We also recall that ∞ H1 (Q, Tp E) is a
Λ-module of rank 1. The idealindΛ (∞ c) = {φ(∞ c) | φ ∈ HomΛ (∞ H1 (Q, Tp E), Λ)}in Λ measures the Λ-divisibility of ∞ c in ∞ H1 (Q, Tp E).
Lemma 1. Let E be an elliptic curve and p an odd prime such that E admits an isogenyof degree p. Then the fine Selmer group Y (E/∞ Q) is a finitely generated Zp -module,i.e. its µ-invariant vanishes.
1 This group is sometimes called the “strict” or “restricted” Selmer group.

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Proof. The extension K of Q fixed by the kernel of ρφ : GQ
 Aut(E[φ]) is a cyclic
extension of degree dividing p − 1. Let ∆ be the Galois group of K/Q. Over the abelianfield K, the curve admits a p-torsion point. We can therefore apply Corollary 3.6in [CS05] (a consequence of the theorem of Ferrero-Washington) to Y (E/∞ K) where∞ K is the cyclotomic Zp -extension of K. This proves that Y (E/∞ K) is a finitely
generated Zp -module. Write R(E/∞ Q) and R(E/∞ K) for the dual of Y (E/∞ Q) andY (E/∞ K) respectively. Then we have the following diagram0 R(E/∞ K)∆f

 H1 (∞ K, Ep∞ )∆f0

 R(E/∞ Q) H1 (∞ Q, Ep∞ )ff
H1 (∆, E(∞ K)p∞ )and since the group ∆ is of order prime to p, the kernel on the right is trivial.
We deduce that the left hand side is injective, too, and hence that the dual map
Y (E/∞ K)
 Y (E/∞ Q) is surjective. Therefore Y (E/∞ Q) is a finitely generated
Zp -module.
Theorem 2.
If E/Q is an elliptic curve without complex multiplication and p > 2 a prime such thatthe presentation ρp : GQ
 Aut(Tp E) is either surjective or that E[p] is reducible
then charΛ (Y (E/∞ Q)) divides indΛ (∞ c).
Proof. If we are in the surjective case, then the theorem is a known consequence ofKato’s Euler system. See Theorem 2.3.3 and Proposition 3.5.8 in [Rub00]. In the lattercase when ρp is not surjective, we know that there exists an isogeny φ : E
 E 0 ofdegree p defined over Q. The Euler system argument gives us only a divisibility of theform˛charΛ (Y (E/∞ Q)) ˛ pt · indΛ (∞ c)for some integer t > 0, see Theorem 2.3.4 in [Rub00]. The previous lemma shows now
that charΛ (Y (E/∞ Q)) is not divisible by p and hence we can take t to be equal to0.

3The Selmer group

Suppose now that the curve E has good and ordinary reduction at the odd prime p.
It is known that there exists an element Lp (E/Q, T ) ∈ Λ ⊗ Qp , called the analytic padic L-function, which interpolates in a certain precise way the Hasse-Weil L-function
associated to E which we are going to recall now. Let γ be a topological generator ofΓ = Gal(∞ Q/Q). Let χ : Γ µp∞ be a Dirichlet character of conductor pk+1 . It isdetermined by its image χ(γ) = ζ which is a primitive root of unity of order pk . Then
Lp (E/Q, T ) is characterised byLp (E/Q, ζ − 1) =

1
αk+1·

pk+1
LE (χ−1 , 1)··.
−1τ (χ )ΩE(1)Here τ (χ−1 ) is the usual Gauss sum and α is the unit root of the characteristic polynomial of Frobenius acting on Tp E. The real Néron period of E is denoted by ΩE andLE (χ−1 , s) is the Hasse-Weil L-function attached to E twisted by the character χ−1 .
We recall from [Ste89] that an elliptic curve E/Q is called optimal among the curvesin the isogeny class of E if the map ϕ∗ : Pic0 (E) Pic0 (X1 (N )) induced by themodular parametrisation ϕ : X1 (N )
 E is injective. It is conjectured that the3

optimal curve is a curve with minimal analytic µ-invariant. It is also conjectured thatthe µ-invariant of the optimal curve is zero. Greenberg and Vatsal have shown inProposition 3.7 of [GV00] that the p-adic L-series of an optimal curve is integral, i.e.
Lp (E/Q, T ) ∈ Λ.
Lemma 3. Let p > 2 be a prime. Let E/Q be an elliptic curve without complexmultiplication with good ordinary reduction at p and such that E[p] is reducible. Suppose
E is the optimal curve in the isogeny class. Then char Λ (X(E/∞ Q)) divides the ideal
Lp (E/Q, T ) · Λ.
Proof. We follow the proof of Theorem 2.3.8 in [Rub00].
Let ∞ Qp be the cyclotomic Zp -extension of Qp . Define the singular local cohomology
group Z(E/∞ Q) = ∞ H1s (Qp , Tp E) to be the dual of E(∞ Qp ) ⊗ Qp /Zp . It is a Λ-moduleof rank 1. By global duality (see Proposition 1.3.2 in [PR95]), we have the following
exact sequence0≺Y (E/∞ Q) ≺X(E/∞ Q) ≺Z(E/∞ Q) ≺1∞ H (Q, Tp E) ≺0. (2)Write ∞ cs for the image of ∞ c in Z(E/∞ Q). Theorem 16.6.2 in [Kat04] states thatthe image of ∞ cs via the Perrin-Riou–Coleman map Col : Z(E/∞ Q)   Λ is up toa p-adic unit the analytic p-adic L-function Lp (E/Q, T ). Here we use that Greenbergand Vatsal [GV00, Theorem 3.1] have shown that the canonical period associated tothe newform corresponding to E differs from ΩE by a p-adic unit, if E is the optimal
curve.
Rohrlich [Roh84] has shown that Lp (E/Q, T ) is non-zero. Hence ∞ cs is not torsion
and the characteristic ideal of the Λ-torsion module Z(E/∞ Q)/∞ cs Λ, which is equalto Col(Z(E/∞ Q))/Lp (E/Q, T ) Λ contains Lp (E/Q, T ) Λ.
The sequence (2) induces an exact sequence of Λ-modules
0≺

Y (E/∞ Q) ≺X(E/∞ Q) ≺Z(E/∞ Q)≺∞ cs Λ1∞ H (Q, Tp E)∞c Λin which all terms are known to be torsion Λ-modules. We know that ∞ H1 (Q, Tp E) is a
Λ-module of rank 1 and hence there is a Λ-morphism ψ from ∞ H1 (Q, Tp E) to Λ whosekernel is Λ-torsion and whose cokernel is pseudo-null. Since ∞ c cannot be torsion, thequotient on the right hand side of the above sequence is Λ-torsion and its characteristic
ideal is contained in ψ(∞ c) Λ. The latter is contained in indΛ (∞ c).
So, using theorem 2, we conclude that“ H1 (Q, T E) ”−1“ Z(E/ Q) ”∞p∞· charΛ
charΛ (X(E/∞ Q)) ⊃ charΛ (Y (E/∞ Q)) · charΛ∞ cs Λ∞c Λ`´−1⊃ indΛ (∞ c) · Lp (E/Q, T ) Λ · indΛ (∞ c)⊃ Lp (E/Q, T ) ΛTheorem 4.
Let E/Q be an elliptic curve without complex multiplication and let p > 2 be a prime.
Suppose that E has good ordinary reduction at p and that the representation ρ p is either
surjective or that E[p] is reducible. Then charΛ (X(E/∞ Q)) divides the ideal generatedby Lp (E/Q, T ).
Proof. If the representation ρp is surjective, then this is Theorem 17.4. of Kato [Kat04].
Suppose now that E[p] is reducible. Then there is an isogeny φ from E to the optimal
curve E opt in the isogeny class of E. Note that (1) and the formula for the changeof the µ-invariant by Perrin-Riou [PR87, Appendice] show that the statement thatcharΛ (X(E/∞ Q)) contains Lp (E/Q, T ) Λ is invariant under isogeny. So the conclusiondrawn for E opt in the previous lemma applies also to E.

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Theorem 5.
The analytic p-adic L-function Lp (E/Q, T ) belongs to Λ for all elliptic curves E/Qwith good ordinary reduction at p > 2.
Proof. If the elliptic curve E admits no isogenies of degree dividing p this is well-knownby [GV00, Proposition 3.7]. If this is not the case, then E[p] is reducible and we haveseen in the previous theorem 4 that the ideal generated by Lp (E/Q, T ) is divisible byan integral ideal charΛ (X(E/∞ Q)).
Corollary 6. If E/Q is a semi-stable elliptic curve and p > 3 a prime of good ordinaryreduction, then charΛ (X(E/∞ Q)) divides the ideal generated by Lp (E/Q, T ).
Proof. By a theorem of Serre ([Ser96, Proposition 1] and [Ser72, Proposition 21]), we
know that the image of the representation ρ̄p : GQ
 Aut(E[p]) is either the whole ofGL2 (Fp ) or it is contained in a Borel subgroup. In the latter case the representation ρp
 Aut(Tp E) is surjective
is reducible and in the first case the representation ρp : GQby another result of Serre [Ser81, Lemme 15].
Unfortunately, the hypothesis that E is semi-stable can not be dropped. There arecurves E/Q such that ρ̄p has its image in the normaliser of a Cartan subgroup. In thiscase there are no p-torsion points defined over an abelian extension of Q. Similar there
are also curves without complex multiplications for p = 5 such that the image of ρ5maps to the exceptional subgroup S4 in PGL(F5 ).
The methods in this article are not sufficient to extend the main theorem 4 to thesecases. The best we can do is the following
Proposition 7. Let E/Q be an elliptic curve without complex multiplication, with goodand ordinary reduction at p > 13 or p = 7. If the conjecture of Iwasawa on the vanishingof the classical µ-invariant in cyclotomic Zp -extensions is valid for abelian extensionsof imaginary quadratic fields, then charΛ (X(E/∞ Q)) divides the ideal generated by
Lp (E/Q, T ).
Proof. By theorem 4, we may assume that the image of ρ̄p is contained in the normaliser of a Cartan subgroup. The case of the exceptional subgroups is excluded by
the hypothesis on p by Lemme 18 in [Ser81].
The idea of the proof is the same as for the proofs of Theorem 2 and Theorem 4, but
we replace the Corollary 3.6 in [CS05] by the previous Corollary 3.5 with L being thefield Q(E[p]). In our case L is an abelian extension of an imaginary quadratic field.

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